International Journal of Applied Mathematics & Statistical Sciences (IJAMSS) ISSN(P): 2319-3972; ISSN(E): 2319-3980 Vol. 3, Issue 2, Mar 2014, 1-10 Š IASET
A NEW HORIZON FOR MULTIPLE ATTRIBUTE GROUP DECISION MAKING PROBLEMS WITH VAGUE SETS A. SOLAIRAJU1, P. JOHN ROBINSON2 & T. LENIN3 1
Associate Professor, Department of Mathematics, Jamal Mohamed College, Tiruchirappalli, Tamil Nadu, India 2
Assistant Professor, Department of Mathematics, Bishop Heber College, Tiruchirappalli, Tamil Nadu, India
3
Associate Professor, Department of Mathematics, Khader Mohideen College, Adirampattinam, Tamil Nadu, India
ABSTRACT Since its entry in the literature, vague set theory has received more and more attention, because many of the real life problems are information in the form of vague values. For Multiple Attribute Group Decision Making (MAGDM) problems where the attribute weights and the expert weights are real numbers and the attribute values take the form of vague values, a new approach is introduced in this paper. The IVOWA operator is introduced and utilized for aggregating the vague information. The induced vague ordered weighted averaging operator (IVOWA) for vague sets is introduced and a MAGDM model is developed based on the IVOWA operator and the vague weighted averaging (VWA) operator. A simple illustration is presented to show the effectiveness of the proposed mode and a comparison of the proposed model is made with an existing method.
KEYWORDS: Multiple Attribute Group Decision Making, Induced Vague Ordered Weighted Averaging Operator (IVOWA), Vague Sets
INTRODUCTION Since the theory of fuzzy sets [20] was proposed in 1965, it has been applied in many uncertain information processing problems successfully, since in the real world there is vague information about different applications. In [8], Gau & Buehrer pointed out the drawback of using the single membership value in fuzzy set theory. In order to tackle this problem, they [8] proposed the notion of Vague Sets (VSs), which allow using interval-based membership instead of using point-based membership as in FSs. The interval-based membership generalization in VSs is more expressive in capturing vagueness of data. However, VSs are shown to be equivalent to that of IFSs [6]. For this reason, the interesting features for handling vague data that are unique to VSs are largely ignored. Atanasov [4, 5] proposed Intuition fuzzy set theory. Gau and Buehrer [8] proposed the concept of Vague set. Bustince & Burillo [6] proposed that Vague set was intuitionistic fuzzy sets and unified the intuition fuzzy set and the Vague set. As the Vague set [6] took the membership degree, non-membership degree and hesitancy degree into account, and has more ability to deal with uncertain information than traditional fuzzy set, lots of scholars pay attentions to the research of Vague set. Atanassov and Gargov [5] extended the intuition vague set and proposed the concept of interval intuition vague set, also named interval Vague set. A Vague Set (VS), as well as an Intuitionistic Fuzzy Set (IFS), is a further generalization of an fuzzy set. Instead of using point-based membership as in FSs, interval-based membership is used in a VS. The interval-based membership in VSs is more expressive in capturing vagueness of data. In the literature, the notions of IFSs and VSs are regarded as www.iaset.us
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A. Solairaju, P. John Robinson & T. Lenin
equivalent, in the sense that an IFS is isomorphic to a VS. Furthermore, due to such equivalence and IFSs being earlier known as a tradition, the interesting features for handling vague data that are unique to VSs are largely ignored. Decision-making is the process of finding the best option from all of the feasible alternatives. Sometimes, decision-making problems considering several criteria are called multi-criteria decision-making (MCDM) problems. The MCDM problems may be divided into two kinds. One is the classical MCDM problems [10, 11], among which the ratings and the weights of criteria are measured in crisp numbers. Another is the fuzzy multiple criteria decision-making (FMCDM) problems, among which the ratings and the weights of criteria evaluated on imprecision and vagueness are usually expressed by linguistic terms, fuzzy numbers or intuition fuzzy numbers. A MAGDM problem is to find a desirable solution from a finite number of feasible alternatives assessed on multiple attributes, both quantitative and qualitative. In order to choose a desirable solution, the decision maker often provides his/her preference information which takes the form of numerical values, such as exact values, interval number values and fuzzy numbers. However, under many conditions, numerical values are inadequate or insufficient to model real-life decision problems. Indeed, human judgments including preference information may be stated in vague information. Hence, MAGDM problems under vague environment [10, 12, 13, 14, 15, 16, 17] is an interesting area of study for researchers in the recent days. Different types of aggregation operators are found in the literature for aggregating the information. A very common aggregation method is the ordered weighted averaging (OWA) operator. It provides a parameterized family of aggregation operators that includes as special cases the maximum, the minimum and the average criteria. Since its appearance, the OWA operator has been used in a wide range of applications. Induced intuitionistic fuzzy operators are already in the literature [18, 19]. In this paper we propose the weighted averaging operator, ordered weighted averaging operator and the induced ordered weighted averaging operator for vague sets. We also propose a general model for decision making utilising these operators for vague sets together with a new distance function defined based on the distance functions from [9].
PRELIMINARIES OF VAGUE SETS In this section, we some basic concepts related to VSs are discussed. Let U be a classical set of objects, called the universe of discourse, where an element of U is denoted by u. Definition (Vague Set) A vague set A in a universe of discourse U is characterized by a true membership function, t A, and a false membership function, fA , as follows [7]: tA : U → [0, 1], fA : U→ [0, 1], and tA (u) + fA (u) ≤ 1, where tA (u) is a lower bound on the grade of membership of u derived from the evidence for u, and f A (u) is a lower bound on the grade of membership of the negation of u derived from the evidence against u. Suppose U = {u1, u2,..., un}. A vague set A of the universe of discourse U can be represented by n
A= [t(u i ),1-f(u i )] / u i , 0 t(u i ) 1-f(u i ) 1,1 i n . i=1
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A New Horizon for Multiple Attribute Group Decision Making Problems with Vague Sets
In other words, the grade of membership of ui is bounded to a subinterval [tA(ui), 1 − fA(ui)] of [0, 1]. Thus, VSs are a generalization of FSs, since the grade of membership μ A(u) of u in the above definition may be inexact in a VS. Basic Definitions and Operations in Vague Sets Let x, y be the two vague values in the universe of discourse U, x = [t x,1 - fx], y = [ty,1 - fy], where tx, fx, ty, fy
[0,1] and tx + fx ≤ 1,ty + fy ≤ 1; the operation and relationship between vague values is defined as follows [1, 2, 3]: Definition The minimum operation of vague values x and y is defined by x
y =[min(tx, ty),min(1 - fx, 1 - fy)]
= [min (tx, ty), 1 – max (1 - fx, 1 - fy)] Definition The maximum operation of vague values x and y is defined by x
y =[max(tx, ty),max(1 - fx, 1 - fy)]
= [max (tx, ty), 1 – min (1 - fx, 1 - fy)] Definition The complement of vague value x is defined by
x = [fx, 1 - tx] Let A, B be two VSs in the universe of discourse U = {u1, u2, . . ., u n}, n
A=
n
[tA (ui),1 - fA (ui)]/ui and B =
i1
[tB (ui),1 – fB (ui)]/ui , then the operations between VSs are defined as
i1
follows. The intersection of VSs A and B is defined by
A
B=
n
{[tA (ui), 1 - fA (ui)]
[tB (ui), 1 – fB (ui)]}/ui
i1
The union of vague sets A and B is defined by A
B=
n
{[tA (ui), 1 - fA (ui)]
[tB (ui), 1 – fB (ui)]}/ui
i1
The complement of vague set A is defined by n
A=
[fA (ui), 1 - tA (ui)]/ui.
i1
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A. Solairaju, P. John Robinson & T. Lenin
Definition For vague value x = [tx, 1 - fx], define the defuzzification function to get the precise value as follows: Dfzz (x) = tx / (tx + fx). Graphical Representation of a VS As we can see that the difference between VSs and IFSs is due to the definition of membership intervals. We have [tA(u), 1 – fA(u)] for u in A (a VS), but < u A(x), vA(x) > for x in A (an IFS). Here the semantics of u A is the same as with tA and vA is the same as with fA. However, the boundary (1 − fA ) is able to indicate the possible existence of a data value, as already mentioned by An Lu and Wilfred [1,2,3]. This subtle difference gives rise to a simpler but meaningful graphical view of data sets. Figure 1 represents the existence of the vague data value in between the boundaries of t A and (1- fA).
Figure 1: Geometrical Interpretation of a Vague Set It can be seen that, the shaded part formed by the boundary in a given VS in Figure 1 naturally represents the possible existence of data. Thus, this “hesitation region” corresponds to the intuition of representing vague data. The choice of the membership boundary also has interesting implications on modelling relationship between vague data. In the following, a new distance function is defined for ranking the alternatives in MAGDM problems. Definition Let A
t A ( xi ), f A ( xi ) , B tB ( xi ), f B ( xi )
be two vague values. Then the Euclidean distance between
A and B is given as follows:
d ( A, B)
1 n (t A ( xi ) tB ( xi ))2 ((1 f A ( xi )) (1 f B ( xi )))2 2 i 1
THE IVOWA OPERATOR FOR GROUP DECISION MAKING In this paper we define a new operator called the Induced Vague Ordered Weighted Averaging (IVOWA) operator for vague sets. Definition Let
a j (t j , f j ), j 1, 2,..., n be a collection of vague values, and let the vague weighted averaging operator
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A New Horizon for Multiple Attribute Group Decision Making Problems with Vague Sets
VWA is defined as: VWA : Q Q if n
n n n VWA (a1 , a 2 ,..., a n ) j a j = 1 (1 t j ) j , (1 f j ) j j 1 j 1 j 1
Where
n
=1 .
(1 , 2 ,..., n )T be the weight vector of a j (t j , f j ) and j 0,
j 1
j
Definition Let
a j (t j , f j ), j 1, 2,..., n be a collection of vague values, then the Vague Ordered Weighted Averaging
(VOWA) operator of dimension n is given by the mapping VOWA : Q Q , with an associated weight vector n
w=(w1 ,w 2 ,..., w n )T such that
wj 0
n
w
and
j
=1 .
j1
n n n w w VOWAw (a1 , a 2 ,..., a n ) w j a ( j ) = 1 (1 t ( j ) ) j , (1 f ( j ) ) j j 1 j 1 j 1
Where
(1), (2),..., (n) is a permutation of (1,2,…,n) such that ( j 1) ( j )
for all j=2,3,…,n.
Definition The IVOWA operator is defined as follows:
IVOWAw
u, a1 , u, a 2 ,..., u, a n
Where w=(w1 ,w 2 ,..., w n )
n n n w w w j g j = 1 (1 t j ) j , (1 f j ) j j 1 j 1 j 1
T
is a weighting vector such that
w j [0,1] and
n
w =1, j 1
j
g j (t j , f j ) is the
a i value of VOWA pair ui , a i having the jth largest ui [0,1] and u i in ui , a i is called as the order inducing variable and
a i is the vague value. The IVOWA operator satisfies the following properties:
Commutativity
IVOWAw Where
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u ,a 1
u , a 1
1
1
, u2 , a 2 ,..., un , a n
, u2 , a2 ,..., un , an
IVOWA u , a w
1
1
, u2 , a2 ,..., un , an
is any permutation of u , a 1
1
, u2 , a 2 ,..., un , a n
.
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A. Solairaju, P. John Robinson & T. Lenin
Idempotency
a j (t j , f j ), a (t , f ) for all j,
If a j a , where Then IVOWAw
u ,a
1
1
a
, u2 , a 2 ,..., un , a n
Monotonicity If a j a j for all j, then
IVOWAw
u ,a 1
1
, u2 , a 2 ,..., un , a n
IVOWA u , a w
1
1
, u2 , a2 ,..., un , an
PROPOSED MODEL OF MAGDM Let
A { A1 , A2 ,..., An } be a set of alternatives, G {G1 , G2 ,..., Gn } be the set of alternatives,
(1 , 2 ,..., n )
is the weighting vector of the attribute G j , j=1,2,…,n, where
j [0,1] ,
n
j 1
j
=1 .
Let D {D1 , D2 ,..., Dt } be the set of decision makers, V (V1 , V2 ,..., Vn ) be the weighting vector of the decision makers, with Vk [0,1],
t
V k 1
k
1 . Let Rk r ij
(k )
mn
tij ( k ) , fij ( k )
mn
be the vague decision matrix, where
tij ( k ) is the
degree of the truth membership value that the alternative Ai satisfies the attribute G j given by the decision maker
Dk and fij ( k ) is the degree of false membership value that the alternative for the alternative Ai , where
tij ( k ) , fij ( k ) [0,1] and, tij ( k ) fij ( k ) 1 , i=1,2,…,m, j=1,2,…,n, k=1,2,…,t The developed model of MAGDM is given as follows: Step 1: Utilize the vague decision matrix
has the associated weighting vector w=(w 1 ,w 2 ,...., w n )
r ij tij , fij IVOWAw V1 , r ij collective decision matrix
Rk r ij
(k )
mn
(1)
Rk r ij
, V2 , r ij
(k )
mn
tij ( k ) , fij ( k )
and the IVOWA operator which
T
(1)
,..., Vt , r ij
(1)
, i=1,2,…,m, j=1,2,…,n, to aggregate into a
, where V (V1 , V2 ,..., Vt ) be the weighting vector of the decision maker.
Step 2: Utilizing the information from the collective decision matrix
mn
Rk r ij
(k )
and the VWA operator mn
r i ti , fi VWA r i1 , r i 2 ,..., r in , i=1,2,…,m, derive the collective overall preference values of the alternative Ai , when
(1 , 2 ,..., n )T is the weighting vector of the attributes.
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A New Horizon for Multiple Attribute Group Decision Making Problems with Vague Sets
Step 3: Calculate the distance between the collective overall preference values
r i and the positive ideal vague
value r , or the negative ideal vague value r , where r = (1,0) and r = (0,1). Using the Euclidean distance function
we can find the distances between the collective overall preference values
r i and the positive ideal vague value r as
follows:
d (r i , r )
1 n (tri ( xi ) t ( xi )) 2 ((1 f ri ( xi )) (1 f ( xi ))) 2 r r 2 i 1
Step 4: Rank all the alternatives Ai , where i = 1, 2,…,m and select the best one in accordance with the distance obtained in step 3.
NUMERICAL ILLUSTRATION Suppose an investment company, wanting to invest a sum of money in the best option, and there is a panel with five possible alternatives to invest the money; A1 is an IT company, A2 is a multinational company, A3 is a tools company, A4 is an airlines company and A5 is an automobile company. The investment company must take a decision according to the four following attributes; G 1 is the risk analysis, G2 is the growth analysis, G3 is the socio-political impact analysis and G4 is the environmental impact analysis. The five possible alternatives Ai , where i = 1,2,…,m, are to be evaluated by three decision makers whose weighting vector is V = (0.35,0.40,0.25)T under the above said four attributes whose weighting vector is = (0.2,0.1,0.3,0.4)T , which gives the decision matrices of vague values
(0.4873, 0.7256) (0.3271, 0.9001) R1 (0.5238, 0.8011) (0.7218, 0.6283) (0.6257, 0.7983)
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Rk r ij
(k )
54
, k 1, 2,3 :
(0.5221, 0.7222) (0.6286, 0.8312) (0.4427, 0.9986) (0.6676, 0.5413) (0.4261, 0.8126) (0.7710, 0.9442) (0.4278, 0.5261) (0.5527, 0.6216) (0.5687, 0.7981) (0.7213, 0.8912) (0.8311, 0.9219) (0.6626, 0.8215) (0.8321,0.9426) (0.6256,0.7119) (0.4136,0.6295)
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A. Solairaju, P. John Robinson & T. Lenin
(0.4351, 0.7846) (0.5121, 0.7221) (0.1009, 0.6221) (0.6321, 0.8221) (0.6226, 0.8108) (0.3009, 0.5129) R 2 (0.5387, 0.9105) (0.4124, 0.7216) (0.5010, 0.7101) (0.7317, 0.8119) (0.5221, 0.8001) (0.2091, 0.4104) (0.5273, 0.6217) (0.3125,0.7278) (0.4728,0.7182)
(0.2217, 0.7184) (0.6225, 0.9105) (0.4491, 0.5426) (0.2101, 0.4110) (0.6210,0.8109)
(0.3198, 0.8279) (0.7726, 0.8901) R3 (0.5201, 0.7287) (0.3247, 0.4821) (0.7351, 0.9113)
(0.6661, 0.7027) (0.7101, 0.9005) (0.6105, 0.9117) (0.5529, 0.7217) (0.4214,0.5005)
(0.4419, 0.9816)
(0.2211, 0.5221)
(0.6245, 0.7815) (0.6216, 0.8225) (0.5821, 0.6286) (0.7117, 0.9211) (0.7139, 0.8148) (0.4212, 0.5334) (0.8001,0.9112) (0.2221,0.6121)
Step 1: Utilizing the decision information given in the matrix
Rk r ij
(k )
54
, k 1, 2,3 and the IVOWA
operator which has the associated weighting vector w = (0.2, 0.35, 0.45) T, we get a collective decision matrix
R k r ij
(k )
54
(0.4229, 0.7718) (0.5921, 0.8805) R (0.5255, 0.7951) (0.6233, 0.6028) (0.6525, 0.7954)
(0.4933, 0.8041) (0.4256, 0.6666) (0.5020, 0.8267) (0.6442, 0.6673) (0.4540, 0.7443) (0.7252, 0.9219) (0.4847, 0.5964) (0.6080, 0.7326) (0.5629, 0.7741) (0.6868, 0.8452) (0.6461, 0.6475) (0.5586, 0.6835) (0.7634,0.8845) (0.4821,0.6764) (0.4651,0.6111)
Step 2: Utilizing the VWA operator, we obtain the collective overall preference values
r i of the alternatives
Ai , where i = 1, 2,…,5.
r1 (0.4637, 0.7622), r 2 (0.6313, 0.8294), r 3 (0.5628, 0.7458) r 4 (0.6133, 0.6699), r 5 (0.5521, 0.6891). Step 3: Calculating the distances between the collective overall preference values
r i and the positive ideal vague
value
r (1,0) . The distances calculated from the following distance function 1 n d (r i , r ) (tri ( xi ) t ( xi )) 2 ((1 f ri ( xi )) (1 f ( xi ))) 2 r r 2 i 1
are given by:
d1 (r1 , r ) 0.6142, d2 (r 2 , r ) 0.5825, d3 (r 3 , r ) 0.5285, d4 (r 4 , r ) 0.4231, d5 (r 5 , r ) 0.4776.
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A New Horizon for Multiple Attribute Group Decision Making Problems with Vague Sets
Step 4: Rank the alternatives based on the shortest distance: A4 < A 5 < A 3 < A 2 < A 1 Hence A4 is the best alternative. Let us consider the replacing of Step-3 with the correlation coefficient proposed in [14] and [17]. Then the ranking order of the alternatives is obtained as follows: A2 > A 4 > A 3 > A 5 > A 1 Hence A2 is the best alternative. From the comparison, it can be observed that there is a change in the ranking of the best alternatives. In the proposed method with a distance function, A4 is the best alternative, and with the replacement of step-3 in the algorithm with methods as in [14] and [17] it can be seen that A 2 is the best alternative
CONCLUSIONS In this paper we have presented the ordered weighted averaging operators in the context of vague set theory. Vague sets can better handle vagueness and uncertainty than intuitionistic fuzzy sets. Initially the vague weighted averaging (VWA) operator was developed and based on VWA operator, the vague ordered weighted averaging (VOWA) operator was introduced. Finally the induced ordered weighted averaging (I-VOWA) operator was developed and a MAGDM model was proposed using all the above proposed operators. The proposed model of MAGDM can be utilised based on any distance function present in the literature.
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